A class of asynchronous parallel nonlinear accelerated overrelaxation methods for the nonlinear complementarity problems

In accordance with the principle of using sufficiently the delayed information, and by making use of the nonlinear multisplitting and the nonlinear relaxation techniques, we present in this paper a class of asynchronous parallel nonlinear multisplitting accelerated overrelaxation (AOR) methods for solving the large sparse nonlinear complementarity problems on the high-speed MIMD multiprocessor systems. These new methods, in particular, include the so-called asynchronous parallel nonlinear multisplitting AOR-Newton method, the asynchronous parallel nonlinear multisplitting AOR-chord method and the asynchronous parallel nonlinear multisplitting AOR-Steffensen method. Under suitable constraints on the nonlinear multisplitting and the relaxation parameters, we establish the local convergence theory of this class of new methods when the Jacobi matrix of the involved nonlinear mapping at the solution point of the nonlinear complementarity problem is an H-matrix.     

The solution of high-order nonlinear ordinary differential equations by Chebyshev Series

By the use of the Chebyshev series, a direct computational method for solving the higher order nonlinear differential equations has been developed in this paper. This method transforms the nonlinear differential equation into the matrix equation, which corresponds to a system of nonlinear algebraic equations with unknown Chebyshev coefficients, via Chebyshev collocation points. The solution of this system yields the Chebyshev coefficients of the solution function. An algorithm for this nonlinear system is also proposed in this paper. The method is valid for both initial-value and boundary-value problems. Several examples are presented to illustrate the accuracy and effectiveness of the method.     

A power method for nonlinear operators

In this article we consider the problem of computing the dominant eigenvalue of the linearization of a nonlinear operator. We define a power method that converges under natural conditions on the nonlinear operator. This nonlinear power method does not require the linearization itself, but only the action of the nonlinear operator on arbitrary functions. We apply this method to investigate the stability of equilibrium solutions of differential equations.     

求解2—D波动方程系数反问题的迭代方法及收敛性

本文针对地震勘探中提出一类重要的２－Ｄ波动方程反演问题，通过定义一个新的非线性算子将２－Ｄ波动方程的反演问题归结为一个新的非线性算子方程，详细讨论了非线性算子的性质，给出了求解反问题的迭代方法，并证明了这种迭代方法的收敛性。     

A modified SQP-filter method for nonlinear complementarity problem

The nonlinear complementarity problem can be reformulated as a nonlinear programming. For solving nonlinear programming, sequential quadratic programming (SQP) type method is very effective. Moreover, filter method, for its good numerical results, are extensively studied to handle nonlinear programming problems recently. In this paper, a modified quadratic subproblem is proposed. Based on it, we employ filter technique to tackle nonlinear complementarity problem. This method has no demand on initial point. The restoration phase, which is always used in traditional filter method, is not needed. Global convergence results of the proposed algorithm are established under suitable conditions. Some numerical results are reported in this paper.     

STABILITY ANALYSIS OF NONLINEAR GALERKIN ＆ GALERKIN METHODS FOR NONLINEAR EVOLUTION EQUATIONS

Abstract. Ogr object in this artlcle is to describe tbe Galerkln scheme and nonlin-eax Galerkin scheme for the approximation of nonlinear evolution equations, and tostudy the stability of these schemes. Spatial discretizatlon can be pedormed by eitherGalerkln spectral method or nonlinear Galerldn spectral method; time discretizatlort isdone hy Euler sin.heine wklch is explicit or implicit in the nonlinear terms. According tothe stability analysis of the above schemes, the stability of nonllneex Galerkln methodis better than that of Galexkln method.     

A tau method for nonlinear dynamical systems

In this work we present a new Tau method for the solution of nonlinear systems of differential equations which are linear in the derivative of highest order and polynomial in the remaining. We avoid the linearization of the problem by associating to it a nonlinear algebraic system and combine a forward substitution with the Tau method. We develop an adaptive step by step version of this alternative nonlinear tau method and we apply it to several nonlinear dynamical systems.     

A New Jacobi Elliptic Function Expansion Method for Solvinga Nonlinear PDE Describing Pulse Narrowing Nonlinear Transmission Lines

In this article, we apply the first elliptic function equation to find a new kind of solutions of nonlinear partial differential equations (PDEs) based on the homogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method. New exact solutions to the Jacobi elliptic functions of a nonlinear PDE describing pulse narrowing nonlinear transmission lines are given with the aid of computer program, e.g. Maple or Mathematica. Based on Kirchhoff's current law and Kirchhoff's voltage law, the given nonlinear PDE has been derived and can be reduced to a nonlinear ordinary differential equation (ODE) using a simple transformation. The given method in this article is straightforward and concise, and can be applied to other nonlinear PDEs in mathematical physics. Further results may be obtained.     

基于Riccati-Bernoulli辅助常微分方程的Davey-Stewartson方程的行波解

Riccati-Bernoulli辅助常微分方程方法可以用来构造非线性偏微分方程的行波解.利用行波变换,将非线性偏微分方程化为非线性常微分方程, 再利用Riccati-Bernoulli方程将非线性常微分方程化为非线性代数方程组, 求解非线性代数方程组就能直接得到非线性偏微分方程的行波解.对Davey-Stewartson方程应用这种方法, 得到了该方程的精确行波解.同时也得到了该方程的一个Backlund变换.所得结果与首次积分法的结果作了比较.Riccati-Bernoulli辅助常微分方程方法是一种简单、有效地求解非线性偏微分方程精确解的方法.     

Generalized Extended Tanh-Function Method for Traveling Wave Solutions of Nonlinear Physical Equations

In this paper, the generalized extended tanh-function method is used for constructing the traveling wave solutions of nonlinear evolution equations. We choose Fisher's equation, the nonlinear schrödinger equation to illustrate the validity and advantages of the method. Many new and more general traveling wave solutions are obtained. Furthermore, this method can also be applied to other nonlinear equations in physics.     

On the numerical solution of stochastic quadratic integral equations via operational matrix method

In this paper, stochastic operational matrix of integration based on delta functions is applied to obtain the numerical solution of linear and nonlinear stochastic quadratic integral equations (SQIEs) that appear in modelling of many real problems. An important advantage of this method is that it dose not need any integration to compute the constant coefficients. Also, this method can be utilized to solve both linear and nonlinear problems. By using stochastic operational matrix of integration together collocation points, solving linear and nonlinear SQIEs converts to solve a nonlinear system of algebraic equations, which can be solved by using Newton's numerical method. Moreover, the error analysis is established by using some theorems. Also, it is proved that the rate of convergence of the suggested method is O(h²). Finally, this method is applied to solve some illustrative examples including linear and nonlinear SQIEs. Numerical experiments confirm the good accuracy and efficiency of the proposed method.     

多约束非线性整数规划的一种改进的算法

多约束非线性整数规划是一类非常重要的问题,非线性背包问题是它的一类特殊而重要的问题.定义在有限整数集上极大化一个可分离非线性函数的多约束最优化问题.这类问题常常用于资源分配、工业生产及计算机网络的最优化模型中,运用一种新的割平面法来求解对偶问题以得到上界,不仅减少了对偶间隙,而且保证了算法的收敛性.利用区域割丢掉某些整数箱子,并把剩下的区域划分为一些整数箱子的并集,以便使拉格朗日松弛问题能有效求解,且使算法在有限步内收敛到最优解.算法把改进的割平面法用于求解对偶问题并与区域分割有效结合解决了多约束非线性背包问题的求解.数值结果表明了改进的割平面方法对对偶搜索更加有效.     

Travelling wave solutions of nonlinear partial equations by using the first integral method

In this paper, the first integral method is employed for constructing the new exact travelling wave solutions of nonlinear partial differential equations. The power of this manageable method is confirmed by applying it for two selected nonlinear partial equations. This approach can also be applied to other systems of nonlinear differential equations.     

一个解不等式约束非线性规划问题的有效方法

本文提出了一个解不等式约束非线性规划问题有效方法．在这个方法中,考虑解一个等价Ｋｕｈｎ－Ｔｕｃｋｅｒ条件的非线性方程组．这个方程组中ＮＣＰ函数的使用消去了对应于不等式约束的Ｌａｇｒａｎｇｅ乘子的非负性．截断牛顿方法被用来解这个非线性方程组．为了保证全局收敛性,一个强健的损失函数被选为寻查函数,同时方法中插入修正最速下降方向．本文证明了方法的分Ｑ－二阶收敛性,同时指出新方法可以有效地解稀疏大规模非线性规划问题。     

A modification of the Adomian decomposition method for a class of nonlinear singular boundary value problems

In this paper, the Adomian decomposition method is modified to solve a class of nonlinear singular boundary value problems which arise as nonlinear normal modal equations in nonlinear conservative vibratory systems. The effectiveness of the modified method is verified by three examples.     

一类非线性波动方程的显式精确解

本文用直接方法和假设的一种结合求出了一类较广泛的非线性波动方程ｕｔｔ－ａ１ｕｘｘ＋ａ２ｕｔ＋ａ３ｕ＋ａ４ｕＳ＾２＋ａ５ｕ＾３＝０的一些显式精确行波解,贱个有重要的非线性数学物理方程,如φ＾４方程,Ｋｌｅｉｎ－Ｇｏｒｄｏｎ方程,Ｓｉｎｅ－Ｇｏｒｄｏｎ方程,及Ｓｉｎｈ－Ｇｏｒｄｏｎ方程的近似,Ｌａｎｄａｕ－Ｇｉｎｚｂｕｒｇ－Ｈｉｇｇｓ方程,Ｄｕｆｆｉｎｇ方程,非线性电报方程等都可作为该方程的特殊情形得     

Numerical solutions of two-dimensional nonlinear integral analysis

In this paper, the approximate solutions for two different type of two-dimensional nonlinear integral equations: two-dimensional nonlinear Volterra-Fredholm integral equations and the nonlinear mixed Volterra-Fredholm integral equations are obtained using the Laguerre wavelet method. To do this, these two-dimensional nonlinear integral equations are transformed into a system of nonlinear algebraic equations in matrix form. By solving these systems, unknown coefficients are obtained. Also, some theorems are proved for convergence analysis.Some numerical examples are presented and results are compared with the analytical solution to demonstrate the validity and applicability of the proposed method.     

Nonlinear Lagrangian Functions and Applications to Semi-Infinite Programs

In this paper a nonlinear penalty method via a nonlinear Lagrangian function is introduced for semi-infinite programs. A convergence result is established which shows that the sequence of optimal values of nonlinear penalty problems converges to that of semi-infinite programs. Moreover a conceptual convergence result of a discretization method with an adaptive scheme for solving semi-infinite programs is established. Preliminary numerical experiments show that better optimal values for some nonlinear semi-infinite programs can be obtained using the nonlinear penalty method.     

Application of Exp-function method to wave solutions of the Sine-Gordon and Ostrovsky equations

In this work, the Exp-function method is employed to find new wave solutions for the Sine-Gordon and Ostrovsky equation. The equations are simplified to the nonlinear partial differential equations and then different types of exact solutions are extracted by this method. It is shown that the Exp-function method is a powerful analytical method for solving other nonlinear equations occurring in nonlinear physical phenomena. Results are presented in contour plots that show the different values of effective parameters on the velocity profiles.     

Third-order iterative methods free from second derivatives for nonlinear equation

In this paper, we suggest and analyze a new two-step iterative method for solving nonlinear equations, which is called the modified Householder method without second derivatives for nonlinear equation. We also prove that the modified method has cubic convergence. Several examples are given to illustrate the efficiency and the performance of the new method. New method can be considered as an alternative to the present cubic convergent methods for solving nonlinear equations.